which implies that any \(\mathcal {F}^M_t \vee \mathcal {F}^R_t\) -martingale is a
\(\mathcal {F}^{M, S}_t\) -martingales.
As a conclusion,
all the \(\mathcal {F}^M_t\) -martingales and all the \(\mathcal {F}^S_t\) -martingales are \(\mathcal {F}^{M, S}_t\) -martingales.
Greater details on this topic can be found in Aksamit and Jeanblanc [1]}, Jeanblanc et al. [2]}
and, more recently, in Calzolari and Torti [3]}.

